Alg2ACP - Solving Quadratic Equations

Solving Quadratic Equations

Algebra 2 ACP

3 Main Methods for Solving Quadratic Equations

• Factoring

• Quadratic Formula

• Taking the Square Root

Method #1 Factoring

If the equation is in standard form, you can factor the equation and then set each factor equal to zero to find the solutions.

Let's try this example

using the FACTORING method

Step 1: Factor

I factored by grouping, but you can use whatever method you like

Step 2: Set each factor equal to 0

Since the whole equation equals zero, either the first factor or the second factor must be zero.

Step 3: Solve for x to find your solutions

So let's recap Method #1 Factoring

Solve Quadratics by Factoring

Step 1: Factor the equation

Step 2: Set each factor equal to zero

Step 3: Solve for x

*Remember, the equation must equal zero before you factor

Now you try one!

Multiple Select

Solve  $5x^2+8x=0$ by factoring. Select ALL the solutions

•  $x=0$  

•  $x=\frac{8}{5}$  

•  $x=-\frac{8}{5}$  

•  $x=\frac{5}{8}$  

•  $x=-1$  

Here is the solution

Method #2 Quadratic Formula

If the quadratic is in standard form and you cannot factor it, use the quadratic formula.

Let's try this example

by using the QUADRATIC FORMULA method

Step 1: Determine a, b, and c

Remember that standard form is

 $ax^2+bx+c=0$  

Step 2: Plug into the quadratic formula

The quadratic formula is

 $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$  

Step 3: Solve for x

Note that if your square root does not simplify to a whole number, you can leave your answer with the $\pm$  

So let's recap Method #2 Quadratic Formula

Solve Quadratics with the Quadratic Formula

Step 1: Find a, b, and c

Step 2: Plug into the quadratic formula
Step 3: Solve for x

Remember, the equation must equal zero before you begin

Now you try one!

Multiple Choice

Solve $-2x^2-5x+2=0$  by using the quadratic formula

•  $x=\frac{-5\pm\sqrt{33}}{4}$  

•  $x=-2$  and  $x=-\frac{1}{2}$  

•  $x=\frac{5\pm\sqrt{41}}{-4}$  

Here is the solution

Method #3 Taking the Square Root

If the quadratic is in vertex form or there is only one x in the equation, you can solve by taking the square root.

Let's try this example

Step 1: Isolate the square

Manipulate the equation so the square (and everything inside) is on one side and everything else is on the other.

Step 2: Take the square root of both sides

Remember to include the  $\pm$  

Step 3: Solve for x

Note that if your square root does not simplify, you can leave the $\pm$ instead of writing two separate solutions

So let's recap Method #3 Taking the Square Root

Solve Quadratics by Taking the Square Root

Step 1: Isolate the square

Step 2: Take the square root of both sides
Step 3: Solve for x

Remember, this method works if there is only 1 x in the equation

Now you try one!

Multiple Choice

Solve $16x^2-9=0$ by taking the square root

•  $x=\pm\frac{3}{4}$  

•  $x=\pm\frac{9}{16}$  

•  $x=\pm\frac{4}{3}$  

•  $x=\pm\frac{16}{9}$  

Here is the solution

How to choose a method for your problem

• If the equation is in vertex form or only has 1 x, use the Taking the Square Root method.

• If the equation is in standard form and is factorable, use the Factoring method.

• Else, use the Quadratic Formula method.

Now go practice!